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Bond Valuation

This set of notes goes over Bond Valuation for Professor Beason's class.
Course

Investments: Debt, Equity And Derivatives (FIN3144)

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Academic year: 2022/2023
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Virginia Polytechnic Institute and State University

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Bond Valuation

Bond valuation is the process of determining the fair price of a bond. The fair price, or market value, of a bond is the present value of the expected cash flows of the bond. The cash flows of a bond include its periodic interest payments, known as the coupon payments, and the return of principal at maturity, known as the face value or par value.

Factors that influence the fair price of a bond include:

● The bond's coupon rate: A bond's coupon rate is the annual interest rate paid on the bond. The higher the coupon rate, the more attractive the bond is to investors, as it offers a higher yield. ● The bond's maturity: A bond's maturity is the length of time until the bond matures and the principal is returned to the investor. The longer the maturity, the higher the risk to the investor, as there is more time for interest rates to change and for the issuer to default on the bond. As a result, longer-term bonds generally have higher yields to compensate for the increased risk. ● The bond's creditworthiness: The creditworthiness of a bond issuer, also known as the issuer's credit rating, is an assessment of the issuer's ability to make timely payments of interest and principal on the bond. A higher credit rating indicates a lower risk of default and, therefore, a lower yield on the bond.

Bond Valuation Formula

The general formula for bond valuation is:

Bond value = sum of the present value of the expected cash flows

The present value of each cash flow is calculated using the following formula:

Present value = cash flow / (1 + r)^n

Where:

● Cash flow is the expected cash flow for a particular period (e. a coupon payment or the return of principal at maturity) ● r is the required rate of return, also known as the discount rate

● n is the number of periods until the cash flow is received

The required rate of return is the minimum rate of return that an investor expects to receive on an investment. It reflects the level of risk associated with the investment and is used to discount the future cash flows to their present value.

Example

Suppose you are considering buying a bond with a face value of $1,000, a coupon rate of 5%, and a maturity of 10 years. The required rate of return on the bond is 6%. How much should you be willing to pay for the bond?

To answer this question, we need to calculate the present value of each cash flow and sum them up to get the bond's value.

First, let's calculate the present value of the annual coupon payments:

There are 10 years until maturity, so there will be 10 coupon payments of $50 ($1,000 x 5%).

The present value of the first coupon payment is: $50 / (1 + 0)^1 = $47.

The present value of the second coupon payment is: $50 / (1 + 0)^2 = $45.

And so on.

The present value of all 10 coupon payments is: $47 + $45 + ... + $26 = $465.

Next, let's calculate the present value of the return of principal at maturity:

The present value of the return of principal at maturity is: $1,000 / (1 + 0)^10 = $674.

Finally, we can sum the present value of the coupon payments and the present value of the return of principal to find the total bond value:

Total bond value = $465 + $674 = $1,

And so on.

The present value of all 5 coupon payments is: $111 + $102 + ... + $87 = $508.

The present value of the return of principal at maturity is: $2,000 / (1 + 0)^5 = $1,424.

Total bond value = $508 + $1,424 = $1,932. Bond Yield

The bond yield is the rate of return received on a bond. It is the bond's annual interest payment divided by the bond's market price.

There are several different types of bond yields, including:

● Coupon yield: The coupon yield is the bond's annual coupon payment divided by the bond's market price. It is also known as the nominal yield. ● Current yield: The current yield is the bond's annual coupon payment divided by the bond's market price. It is a measure of the bond's income in relation to its market price and does not take into account the time value of money. ● Yield to maturity (YTM): The yield to maturity is the total return anticipated on a bond if the bond is held until it matures. It takes into account the bond's coupon payments, the gain or loss on the bond due to changes in the market interest rate, and the return of principal at maturity. The yield to maturity is the discount rate that equates the bond's present value to its price. ● Yield to call (YTC): The yield to call is the total return anticipated on a bond if the bond is called by the issuer prior to maturity. It takes into account the bond's coupon payments, the gain or loss on the bond due to changes in the market interest rate, and the return of principal at the call date. The yield to call is the discount rate that equates the bond's present value to its price assuming the bond is called on the first call date.

Bond Yield Formula

The general formula for calculating the yield to maturity of a bond is:

YTM = (C + (F - P) / N) / ((F + P) / 2)

Where:

● C is the bond's annual coupon payment ● F is the bond's face value or par value ● P is the bond's market price ● N is the number of years until maturity

The yield to call can be calculated using a similar formula, with the call date substituted for the maturity date.

Example

Suppose you have a bond with a face value of $1,000, a coupon rate of 5%, and a maturity of 10 years. The bond's market price is $950. Calculate the bond's yield to maturity.

To solve this problem, we can use the YTM formula:

YTM = (C + (F - P) / N) / ((F + P) / 2)

Plugging in the values:

YTM = (50 + (1,000 - 950) / 10) / ((1,000 + 950) / 2)

YTM = (50 + 50 / 10) / (1,000 + 950) / 2

YTM = (50 + 5) / (1,950 / 2)

YTM = 55 / 975

YTM = 0 or 5%

Practice Problems

  1. A bond has a face value of $1,000, a coupon rate of 8%, and a maturity of 20 years. The bond's market price is $900. Calculate the bond's yield to maturity.

YTC = 0 or 11%

Bond Pricing and Yield Inverse Relationship

It is important to note that bond prices and bond yields have an inverse relationship. When bond prices increase, bond yields decrease, and vice versa. This is because bond prices and bond yields are inversely related to the required rate of return. When the required rate of return increases, the present value of the expected cash flows decreases, leading to a lower bond price. Conversely, when the required rate of return decreases, the present value of the expected cash flows increases, leading to a higher bond price.

Example

Suppose you have a bond with a face value of $1,000, a coupon rate of 5%, and a maturity of 10 years. The bond's yield to maturity is currently 6%. If the required rate of return on the bond increases to 7%, what will happen to the bond's price?

To solve this problem, we can use the bond valuation formula:

Bond value = sum of the present value of the expected cash flows

First, let's calculate the present value of the annual coupon payments:

There are 10 years until maturity, so there will be 10 coupon payments of $50 ($1,000 x 5%).

The present value of the first coupon payment is: $50 / (1 + 0)^1 = $47.

The present value of the second coupon payment is: $50 / (1 + 0)^2 = $45.

And so on.

The present value of all 10 coupon payments is: $47 + $45 + ... + $26 = $465.

Next, let's calculate the present value of the return of principal at maturity:

The present value of the return of principal at maturity is: $1,000 / (1 + 0)^10 = $674.

Finally, we can sum the present value of the coupon payments and the present value of the return of principal to find the bond's value at a yield to maturity of 6%:

Total bond value = $465 + $674 = $1,

Now, let's see what happens to the bond's value when the required rate of return increases to 7%.

The present value of the first coupon payment is: $50 / (1 + 0)^

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Bond Valuation

Course: Investments: Debt, Equity And Derivatives (FIN3144)

11 Documents
Students shared 11 documents in this course

University: Virginia Polytechnic Institute and State University

Was this document helpful?
Bond Valuation
Bond valuation is the process of determining the fair price of a bond. The fair price, or
market value, of a bond is the present value of the expected cash flows of the bond. The
cash flows of a bond include its periodic interest payments, known as the coupon
payments, and the return of principal at maturity, known as the face value or par value.
Factors that influence the fair price of a bond include:
●The bond's coupon rate: A bond's coupon rate is the annual interest rate paid on
the bond. The higher the coupon rate, the more attractive the bond is to investors,
as it offers a higher yield.
●The bond's maturity: A bond's maturity is the length of time until the bond
matures and the principal is returned to the investor. The longer the maturity, the
higher the risk to the investor, as there is more time for interest rates to change
and for the issuer to default on the bond. As a result, longer-term bonds generally
have higher yields to compensate for the increased risk.
●The bond's creditworthiness: The creditworthiness of a bond issuer, also known
as the issuer's credit rating, is an assessment of the issuer's ability to make
timely payments of interest and principal on the bond. A higher credit rating
indicates a lower risk of default and, therefore, a lower yield on the bond.
Bond Valuation Formula
The general formula for bond valuation is:
Bond value = sum of the present value of the expected cash flows
The present value of each cash flow is calculated using the following formula:
Present value = cash flow / (1 + r)^n
Where:
●Cash flow is the expected cash flow for a particular period (e.g. a coupon
payment or the return of principal at maturity)
●r is the required rate of return, also known as the discount rate

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